shared_sensitivity
plain-language theorem explainer
The shared sensitivity result shows that Jcost(r) is strictly positive for every positive real r different from 1. Researchers modeling equilibria across game theory, efficient markets, and biological homeostasis cite it to confirm that the quadratic response to a perturbation is identical in all three settings. The argument is a direct one-line application of the core positivity lemma for the J-cost function.
Claim. For any real number $r > 0$ with $r ≠ 1$, the J-cost satisfies $Jcost(r) > 0$, where Jcost is the cost function derived from the J-functional.
background
The J-cost function is defined in the Cost module and satisfies Jcost(x) > 0 whenever x > 0 and x ≠ 1. The module JEquilibriumUniversality establishes that the single condition Jcost(1) = 0 serves as the equilibrium fixed point simultaneously for Nash equilibrium, market equilibrium, and health equilibrium. Upstream lemmas in Cost, JcostCore, and LocalCache each prove the same positivity statement by rewriting Jcost as a square divided by x and observing that the numerator is positive when x ≠ 1.
proof idea
This is a one-line wrapper that applies the lemma Jcost_pos_of_ne_one, passing the hypotheses 0 < r and r ≠ 1 directly to obtain the strict inequality.
why it matters
The declaration supplies the shared perturbation sensitivity required by the jEquilibriumUniversalityCert structure, which bundles the three equilibrium propositions together with the statement that all three are identical. It realizes the module claim that a perturbation analysis performed in one domain yields the same multiplier in the others. Within the Recognition Science framework it supports the universality of the J-functional across the J-uniqueness step and the cross-domain consequences of the Recognition Composition Law.
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